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I'm asked to sketch the phase portrait of a symmetric potential well and I had a few questions regarding the behavior of the phase portrait. Would the outer edges of the phase portrait behave similar to how is shown in my sketch as the solid line where they circle the two centers or would they deviate asymptotically along what would be the x-axis depicted by the dashed lines? Another question I had was if damping existed in the potential function and the centers were no longer neutrally stable, what would determine which unstable ends of the saddle point would be drawn to which center?

Phase Portrait Sketch

Any input is appreciated

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No, the dashed lines are not "physical", the solid lines are fine. Your solutions are bounded by their energy levels, that is, you can draw a horizontal line of the energy level in the graph of the potential energy, and the crossings define the extremal points between which the solution oscillates.

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